Recall that a graph is bipartite if its vertex set admits a partition into two classes in such a way that every edge has its ends in two different classes [1]}, [2]}. A complete bipartite graph [2]} is a bipartite graph in which every two vertices from two different classes are adjacent. A task swapping graph of complete bipartite topology [4]}, [5]} is a complete bipartite task swapping graph, where agents are grouped in two layers or classes (the upper and lower) in such a way that agents between different layers are fully-connected, while agents in the same layer are not directly connected. The agents at the upper layer have a centralized control over the agents at the lower layer in a distributed manner in this topology. We assume that each agent in a task swapping graph of complete bipartite topology is labeled in ascending order from the upper left to bottom right as shown in Fig. REF . A task swapping graph of complete bipartite topology having \(n\)  agents with their task assignment represented by permutation \(p \in \mathfrak {S}_n\)  is denoted as \(\Gamma _p^{B(n, k)}\) , where \(k\)  is the number of agents at the upper layer in \(\Gamma _p^{B(n, k)}\) . We call \(k\)  as bipartite index. As with other task swapping graphs, we assume \(n \ge 3\)  for \(\Gamma _p^{B(n, k)}\) . We also assume \(1 \le k <n\)  for \(\Gamma _p^{B(n, k)}\) . Note that \(\Gamma _p^{B(n, k)}\)  is simply \(\Gamma _p^{S(n)}\)  when bipartite index \(k\)  is 1. For example, the task swapping graph of Fig. REF (a) is denoted as \(\Gamma _{p_1}^{B(8, 3)}\)  and the task swapping graph of Fig. REF (b) is denoted as \(\Gamma _{p_2}^{B(8, 3)}\) , respectively, where \(p_1=3\,7\,1\,6\,5\,4\,8\,2 \in \mathfrak {S}_8\)  and \(p_2=2\,4\,7\,8\,6\,1\,3\,5 \in \mathfrak {S}_8\) . We see that a right multiplication of permutation \(p_1 \in \mathfrak {S}_8\)  by transposition \((1\,4)\)  represents an adjacent task swapping between agent 1 and agent 4 in \(\Gamma _{p_1}^{B(8, 3)}\) , while a right multiplication of permutation \(p_1 \in \mathfrak {S}_8\)  by transposition \((1\,2)\)  does not represent an adjacent task swapping in \(\Gamma _{p_1}^{B(8, 3)}\) . Therefore, the generating set \(S_5^\prime =\lbrace (i\,j):1 \le i \le 3 <j \le 8\rbrace \)  is required for finding a minimum-length sequence of adjacent task swappings needed from \(\Gamma _{p_1}^{B(8, 3)}\)  to reach \(\Gamma _{p_2}^{B(8, 3)}\) .
